A general solution and its derivative to
(\ref{eqn:scalarwavenorm}) in the form of standing waves is,
\begin{align}
\begin{split}
N(n)=N^e_\gamma\cosh(\gamma_nn)+N^o_\gamma\sinh(\gamma_nn)\\=N^e_k\cos(k_nn)+N^o_k\sin(k_nn)
\end{split}\label{eqn:scalarwavenormsolnstand}
\end{align}
\begin{align}
\begin{split}
\frac{dN(n)}{dn}=\gamma_n(N^e_\gamma\sinh(\gamma_nn)+N^o_\gamma\cosh(\gamma_nn))\\=-k_n(N^e_k\sin(k_nn)-N^o_k\cos(k_nn))
\end{split}\label{eqn:scalarwavenormsolnstandd}
\end{align}
where $N^e$ and $N^o$ are the complex coefficients of the even and
odd functions. By using the condition of (\ref{eqn:rbgkpos}), and
substituting into (\ref{eqn:scalarwavenormsolnstand}) and
(\ref{eqn:scalarwavenormsolnstandd}), yields the following set of
equations,
\begin{align}
    N^e_\gamma\cos(k_nn)+jN^o_\gamma\sin(k_nn)&=N^e_k\cos(k_nn)+N^o_k\sin(k_nn)\label{eqn:constfcnn}\\
    N^e_\gamma\sin(k_nn)-jN^o_\gamma\cos(k_nn)&=N^e_k\sin(k_nn)-N^o_k\cos(k_nn)\label{eqn:constdern}
\end{align}
Solving for $N^e_\gamma$ in (\ref{eqn:constfcnn}) and
(\ref{eqn:constdern}) and equating yields,
\begin{equation}\label{eqn:constfcndern}
   -jN^o_\gamma\frac{\sin(k_nn)}{\cos(k_nn)}+N^o_k\frac{\sin(k_nn)}{\cos(k_nn)}=jN^o_\gamma\frac{cos(k_nn)}{sin(k_nn)}-N^o_k\frac{\cos(k_nn)}{sin(k_nn)}
\end{equation}
Multiplying both sides of (\ref{eqn:constfcndern}) by
$\cos(k_nn)\sin(k_nn)$ and rearranging gives,
\begin{equation}\label{eqn:constfcndern2}
   N^o_k(sin^2(k_nn)+\cos^2(k_nn))=jN^o_\gamma(\sin^2(k_nn)+\cos^2(k_nn))
\end{equation}
Equation (\ref{eqn:constfcndern2}) simplifies to the following,
\begin{equation}\label{eqn:relNgoNko}
   \boxed{N^o_k=jN^o_\gamma\Longleftrightarrow N^o_\gamma=-jN^o_k}
\end{equation}
Substituting (\ref{eqn:relNgoNko}) into either
(\ref{eqn:constfcnn}) or (\ref{eqn:constdern}) yields,
\begin{equation}\label{eqn:relNgeNke}
   \boxed{N^e_k=N^e_\gamma}
\end{equation}
Finally (\ref{eqn:scalarwavenormsolnstand}) and
(\ref{eqn:scalarwavenormsolnstandd}) can be rewritten in terms of
common coefficients by letting $N^e=N_k^e$ and $N^o=N_k^o$,
\begin{subequations}
\begin{align}
    N(n)&=N^e\cosh(\gamma_nn)-jN^o\sinh(\gamma_nn)\notag\\
    &=N^e\cos(k_nn)+N^o\sin(k_nn)\label{eqn:scalarwavenormsolnstand2}\\
    \frac{dN(n)}{dn}&=\gamma_n(N^e\sinh(\gamma_nn)-jN^o\cosh(\gamma_nn))\notag\\
    &=-k_n(N^e\sin(k_nn)-N^o\cos(k_nn))\label{eqn:scalarwavenormsolnstandd2}
\end{align}
\end{subequations}
or by using an alternate definition by letting $N^e=N_\gamma^e$ and $N^o=N_\gamma^o$,
\begin{subequations}
\begin{align}
    N(n)&=N^e\cosh(\gamma_nn)+N^o\sinh(\gamma_nn)\label{eqn:scalarwavenormsolnstand3a}\\
    &=N^e\cos(k_nn)+jN^o\sin(k_nn)\label{eqn:scalarwavenormsolnstand3b}
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
    \frac{dN(n)}{dn}&=\gamma_n(N^e\sinh(\gamma_nn)+N^o\cosh(\gamma_nn))\label{eqn:scalarwavenormsolnstandd3a}\\
    &=-k_n(N^e\sin(k_nn)-jN^o\cos(k_nn))\label{eqn:scalarwavenormsolnstandd3b}
\end{align}
\end{subequations}
